Deep learning methodology has revealed some major surprises from the perspective of statistical complexity: even without any explicit effort to control model complexity, these methods find prediction rules that give a near-perfect fit to noisy training data and yet exhibit excellent prediction performance in practice. We investigate this phenomenon of ‘benign overfitting’ in the setting of linear prediction, and give a characterization of linear regression problems for which the minimum norm interpolating prediction rule has near-optimal prediction accuracy. The characterization shows that overparameterization is essential: the number of directions in parameter space that are unimportant for prediction must significantly exceed the sample size. We discuss implications for deep networks, for robustness to adversarial examples, and for the rich variety of possible behaviors of excess risk as a function of dimension, and we describe extensions to ridge regression and barriers to analyzing benign overfitting based on model-dependent generalization bounds.

Joint work with Phil Long, Gábor Lugosi, and Alex Tsigler.

Peter Bartlett is a professor in the Division of Computer Science and the Department of Statistics. He is the co-author of the book Learning in Neural Networks: Theoretical Foundations. He has served as associate editor of the journals Machine Learning, Mathematics of Control Signals and Systems, the Journal of Machine Learning Research, the Journal of Artificial Intelligence Research, and the IEEE Transactions on Information Theory. He was awarded the Malcolm McIntosh Prize for Physical Scientist of the Year in Australia for his work in statistical learning theory. He was a Miller Institute Visiting Research Professor in Statistics and Computer Science at U.C. Berkeley in Fall 2001, and a fellow, senior fellow and professor in the Research School of Information Sciences and Engineering at the Australian National University's Institute for Advanced Studies (1993-2003). He is also an honorary professor in the Department of Computer Science and Electrical Engineering at the University of Queensland.